Question

Give an O(n2 ) algorithm (Pseudocode) that, given a sequence S, finds the longest subsequence that first increases then decreases.

For instance, in the sequence S = [10, 4, 5, 11, 2, 7, 4, 3, 9] the longest such subsequence is [4,5,11,7,4,3]. The subsequence does not have to be consecutive. (Hint: Use two arrays, one for increasing subsequences and the other for decreasing subsequences.)

Answer 1

function longestBitonicSequence( Array arr[], Integer n as size )

{

Integer i, j;

/* Allocate memory for LIS[] and initialize LIS values as 1 for

all indexes */

Array lis[n]

for i = 0 to n do

lis[i] = 1

end loop

/* Compute LIS values from left to right */

for i 1 to n-1 do

for j 0 to i-1 do

if (arr[i] > arr[j] && lis[i] < lis[j] + 1) then

lis[i] = lis[j] + 1

end if

end loop

end loop

/* Allocate memory for lds and initialize LDS values for

all indexes */

array lds[n];

for i = 0 to n do

lds[i] = 1

end loop

/* Compute LDS values from right to left */

for i = n-2 to 0 do

for j = n-1 to i+1 do

if (arr[i] > arr[j] && lds[i] < lds[j] + 1) then

lds[i] = lds[j] + 1

end if

end loop

end loop

/* Return the maximum value of lis[i] + lds[i] - 1*/

max = lis[0] + lds[0] - 1;

for i = 1 to n do

if (lis[i] + lds[i] - 1 > max) then

max = lis[i] + lds[i] - 1

end if

end loop

return max;

}